dy/dx+(x+(1+3x²)/(1+x+x³))*y=x³+2x+x²*(1+3x²)/(1+x+x³) y(0)=1

可以使用深度学习框架进行神经网络的构建和训练，这里以PyTorch为例。

首先，定义神经网络模型：

import torch
import torch.nn as nn
import torch.optim as optim

class Net(nn.Module):
    def __init__(self):
        super(Net, self).__init__()
        self.linear1 = nn.Linear(2, 10) # 输入层到隐层
        self.linear2 = nn.Linear(10, 1) # 隐层到输出层

    def forward(self, x):
        x = torch.sigmoid(self.linear1(x))
        x = self.linear2(x)
        return x

接下来，定义训练函数：

def train(net, criterion, optimizer, x_train, y_train):
    net.train()
    running_loss = 0.0
    for i in range(len(x_train)):
        x = x_train[i]
        y = y_train[i]

        optimizer.zero_grad()

        outputs = net(x)
        loss = criterion(outputs, y)
        loss.backward()
        optimizer.step()

        running_loss += loss.item()

    return running_loss / len(x_train)

然后，定义测试函数：

def test(net, criterion, x_test, y_test):
    net.eval()
    running_loss = 0.0
    with torch.no_grad():
        for i in range(len(x_test)):
            x = x_test[i]
            y = y_test[i]

            outputs = net(x)
            loss = criterion(outputs, y)

            running_loss += loss.item()

    return running_loss / len(x_test)

接下来，进行数据处理，将微分方程转化为数据集：

import numpy as np

# 定义微分方程
def f(x, y):
    return x ** 3 + 2 * x + x ** 2 * (1 + 3 * x ** 2) / (1 + x + x ** 3) - (x + (1 + 3 * x ** 2) / (1 + x + x ** 3)) * y

# 定义数据集
x_train = np.linspace(0, 1, 100)
y_train = np.zeros((100, 1))
y_train[0] = 1

for i in range(1, len(x_train)):
    h = x_train[i] - x_train[i - 1]
    k1 = f(x_train[i - 1], y_train[i - 1])
    k2 = f(x_train[i - 1] + h / 2, y_train[i - 1] + h / 2 * k1)
    k3 = f(x_train[i - 1] + h / 2, y_train[i - 1] + h / 2 * k2)
    k4 = f(x_train[i], y_train[i - 1] + h * k3)
    y_train[i] = y_train[i - 1] + h / 6 * (k1 + 2 * k2 + 2 * k3 + k4)

最后，进行神经网络的训练和测试：

# 初始化神经网络模型、损失函数和优化器
net = Net()
criterion = nn.MSELoss()
optimizer = optim.Adam(net.parameters(), lr=0.01)

# 进行训练和测试
for epoch in range(1000):
    train_loss = train(net, criterion, optimizer, x_train, y_train)
    test_loss = test(net, criterion, x_train, y_train)

    if epoch % 100 == 0:
        print('Epoch: {}, Train Loss: {:.6f}, Test Loss: {:.6f}'.format(epoch+1, train_loss, test_loss))

最终，可以得到训练好的神经网络模型，将新的输入数据输入到模型中，即可得到相应的输出。